Module 5 Slides

School of Electrical and Computer Engineering

Control of Mobile Robots

Dr. Magnus Egerstedt Professor School of Electrical and Computer Engineering

Module 5 Hybrid Systems

How make mobile robots move in effective, safe, predictable, and collaborative ways using modern control theory?

So far, the models stay the same over time

We have a designed one-size-fits-all controllers

Lecture 5.1 Switches Everywhere

Magnus Egerstedt, Control of Mobile Robots, Georgia Ins

Switches by Necessity


Switches by Design


Models? Stability and Performance? Compositionality? Traps?

Issues


We need to be able to describe systems that contain both the continuous dynamics and the discrete switch logic

Hybrid Automata = Finite state machines (discrete logic) on steroids (continuous dynamics)

Lecture 5.2 Hybrid Automata


Let, as before, the (continuous) state of the system be x As we will be switching between different modes of operation,

lets add an additional discrete state q Dynamics:

The transitions between different discrete modes can be encoded in a state machine:

Modes, Transitions, Guards, and Resets


The conditions under which a transition occurs are called guard conditions, i.e., a transition occurs from q to q if

As a final component, we would like to allow for abrupt changes in the continuous state as the transitions occur, which we will call resets:

Modes, Transitions, Guards, and Resets


Putting all of this together yields a very rich model known as a hybrid automata (HA) model:

The Hybrid Automata Model


HA Example 1 - Thermostat


HA Example 2 Gear Shift


HA Example 3 Behaviors


What can possibly go wrong when you start switching between different controllers?

Lecture 5.3 A Counter Example


Two modes:

Both modes are asymptotically stable!

A Simple 2-Mode System


Mode 1


Mode 2


HA 1


HA 2


By combining stable modes, the resulting hybrid system may be unstable!

By combining unstable modes, the resulting hybrid system may be stable!

Design stable modes but be aware that this is a risk one may face!

Punchlines


Stable subsystems do not guarantee a stable hybrid system

Lecture 5.4 Danger, Beware!


We saw last time that it was possible to destabilize stable subsystems by an unfortunate series of switches

Ignoring resets, we can write the hybrid system as a switched system:

The switch signal dictates which discrete mode the system is in

Switched Systems


Given a switched system universal, asymptotic stability:

existential, asymptotic stability:

If the switch signal is generated by an underlying hybrid automaton: hybrid, asymptotic stability:

Different Kinds of Stability


Some Results


Practically Speaking


Lets model a ball bouncing on a surface:

Equations of motion in-between bounces:

Bounces:

Lecture 5.5 The Bouncing Ball


The Ball HA


Solving for the Output


Time In-Between Bounces?


Accumulated Bounce Times


So What?


Problem with the bouncing ball: Infinitely many switches in finite time

This is bad: Simulations crash Model is not accurate System behavior fundamentally ill-defined beyond Zeno

point

Lecture 5.6 The Zeno Phenomenon


The Zeno Phenomenon


Example


Super-Zeno?


Zeno is a problem Type 1 is not only detectable, but one can deal with it in a

rather straightforward manner Type 2 is overall hard to handle!

Good News and Bad News


It is clear what should happen! How do we make that mathematically sound? Sliding Mode Control

Lecture 5.7 Sliding Mode Control


Both vector fields point inwards = bad! We should keep sliding along the switching surface Sliding Mode Control

Switching Surfaces

Magnus Egerstedt, Control of Mobile Robots, Georgia Ins 0

g(x) < 0

g(x) = 0

switching surface

Sliding?


g(x) < 0

g(x) = 0

g

x

T

f1

f2

Sliding occurs if g

xf1 < 0 AND

g

xf2 > 0

derivative of g in direction f = Lfg = Lie derivative

Sliding?


g(x) < 0

g(x) = 0

g

x

T

f1

f2

Sliding occurs if

derivative of g in direction f = Lfg

Lf1g < 0 AND Lf2g > 0

= Lie derivative

Next time: But what happens beyond the Zeno point?

A Test For Type 1 Zeno


How do we move beyond the Zeno point?

Lecture 5.8 Regularizations


g(x) < 0

g(x) = 0

f2

f1

x = f1(x)

x = f2(x)

g(x) < 0g(x) 0

Sliding occurs if

Lf1g < 0 AND Lf2g > 0

The Sliding Mode


Back to the Example


The Induced Mode


Regularizations of Type 1 Zeno HA


Regularizations of Type 1 Zeno HA


x =1

Lf2g Lf1gLf2gf1 Lf1gf2

g < 0g > 0

g = 0 and Lf1g < 0 and Lf2g > 0

g = 0 and Lf1g < 0 and Lf2g > 0

We have Models Stability Awareness Zeno Regularizations

Next Module: Back to ROBOTICS!

Hybrid Systems: In Summary


Module 5 Slides

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